Predicting the ultimate supremum of a stable L\'{e}vy process with no negative jumps

Given a stable L'{e}vy process $X=(X_t{)}_{0\le t\le T}$ of index $\alpha \in (1,2)$ with no negative jumps, and letting $S_t=\sup_{0\le s\le t}X_s$denoteitsrunningsupremumfor$t\in [0,T]$, we consider the optimal prediction problem $$V=\underset{0\le \tau \le T}{\inf }\mathsf{E}(S_T-X_{\tau }{)}^p,$$ where the infimum is taken over all stopping times $\tau$ of $X$, and the error parameter $p\in (1,\alpha )$ is given and fixed. Reducing the optimal prediction problem to a fractional free-boundary problem of Riemann–Liouville type, and finding an explicit solution to the latter, we show that there exists ${\alpha }_{\ast }\in (1,2)$ (equal to 1.57 approximately) and a strictly increasing function $p_{\ast }:({\alpha }_{\ast },2)\to (1,2)$ satisfying $p_{\ast }({\alpha }_{\ast }+)=1$, $p_{\ast }(2-)=2$ and $p_{\ast }(\alpha )<\alpha$ for $\alpha \in ({\alpha }_{\ast },2)$ such that for every $\alpha \in ({\alpha }_{\ast },2)$ and $p\in (1,p_{\ast }(\alpha ))$ the following stopping time is optimal $${\tau }_{\ast }=\inf \{t\in [0,T]:S_t-X_t\ge z_{\ast }(T-t{)}^{1\mathrm{/}\alpha }\},$$ where $z_{\ast }\in (0,\mathrm{\infty })$ is the unique root to a transcendental equation (with parameters $\alpha$ and $p$). Moreover, if either $\alpha \in (1,{\alpha }_{\ast })$ or $p\in (p_{\ast }(\alpha ),\alpha )$ then it is not optimal to stop at $t\in [0,T)$ when $S_t-X_t$ is sufficiently large. The existence of the breakdown points ${\alpha }_{\ast }$ and $p_{\ast }(\alpha )$ stands in sharp contrast with the Brownian motion case (formally corresponding to $\alpha =2$), and the phenomenon itself may be attributed to the interplay between the jump structure (admitting a transition from lighter to heavier tails) and the individual preferences (represented by the error parameter $p$).